In this paper, we studied the nucleation process of a vacuum bubble in the Kerr-AdS\({}_5\)  spacetime. It is an extension of the cosmological scenario [1]}, [2]}, [3]}, in which a quasi-dS\({}_4\)  expansion is realized on the nucleated bubble in an AdS\({}_5\)  spacetime, to a situation where a gravitational impurity, i.e., a Kerr-AdS\({}_5\)  black hole, is involved in the false vacuum state. It is known that an AdS spacetime is non-linearly unstable and small black holes may form in the AdS due to the gravitational turbulent instability [4]}, [5]}. We then found that the cosmological scenario works even in the less symmetric background, i.e., Kerr-AdS\({}_5\)  false vacuum state, without contradicting the de Sitter swampland conjectures.
